How do you derive and interpret a circuit's transfer function?

A transfer function H(s) is the ratio of output to input in the s-domain: H(s) = Vout(s)/Vin(s), assuming zero initial conditions. Derivation involves writing the circuit equations in s-domain, solving for output/input ratio, and expressing as a ratio of polynomials. Poles (denominator roots) determine stability and natural response modes; zeros (numerator roots) affect frequency response. Transfer functions enable frequency response analysis, filter design, and control system integration.